CAT (QA) PYQs

P, Q, R and S are four towns. One can travel between P and Q along 3 direct paths, between Q and S along 4 direct paths, and between P and R along 4 direct paths. There is no direct path between P and S, while there are few direct paths between Q and R, and between R and S. One can travel from P to S either via Q, or via R, or via Q followed by R, respectively, in exactly 62 possible ways. One can also travel from Q to R either directly, or via P, or via S, in exactly 27 possible ways. Then, the number of direct paths between Q and R is

In how many ways can $8$ identical pens be distributed among Amal, Bimal, and Kamal so that Amal gets at least $1$ pen, Bimal gets at least $2$ pens, and Kamal gets at least $3$ pens?

With rectangular axes of coordinates, the number of paths from $(1,1)$ to $(8,10)$ via $(4,6)$, where each step from any point $(x, y)$ is either to $(x$, $y+1)$ or to ( $x+1, y$ ), is